Three-dimensional wedge filling in ordered and disordered systems

We investigate interfacial structural and fluctuation effects occurring at continuous filling transitions in 3D wedge geometries. We show that fluctuation-induced wedge covariance relations that have been reported recently for 2D filling and wetting have mean-field or classical analogues that apply to higher-dimensional systems. Classical wedge covariance emerges from analysis of filling in shallow wedges based on a simple interfacial Hamiltonian model and is supported by detailed numerical investigations of filling within a more microscopic Landau-like density functional theory. Evidence is presented that classical wedge covariance is also obeyed for filling in more acute wedges in the asymptotic critical regime. For sufficiently short-ranged forces mean-field predictions for the filling critical exponents and covariance are destroyed by pseudo-one-dimensional interfacial fluctuations. We argue that in this filling fluctuation regime the critical exponents describing the divergence of length scales are related to values of the interfacial wandering exponent ζ(d) defined for planar interfaces in (bulk) two-dimensional (d = 2) and three-dimensional (d = 3) systems. For the interfacial height lw ∼ θ-α)-βw, with θ the contact angle and α the wedge tilt angle, we find βw = ζ(2)/2(1-ζ(3)). For pure systems (thermal disorder) we recover the known result βw = 1/4 predicted by interfacial Hamiltonian studies whilst for random-bond disorder we predict the universal critical exponent β ∼ even in the presence of dispersion forces. We revisit the transfer matrix theory of three-dimensional filling based on an effective interfacial Hamiltonian model and discuss the interplay between breather, tilt and torsional interfacial fluctuations. We show that the coupling of the modes allows the problem to be mapped onto a quantum mechanical problem as conjectured by previous authors. The form of the interfacial height probability distribution function predicted by the transfer matrix approach is shown to be consistent with scaling and thermodynamic requirements for distances close to and far from the wedge bottom respectively